Logarithmics Differentiation Help
We next show how to use the logarithm as an aid to differentiation. The key idea is that if F is a function taking positive values then we can exploit the formula
Calculate the derivative of the function
F ( x ) = (cos x ) (sin x ) , 0 < x < .
We take the natural logarithm of both sides:
Now we calculate the derivative using the formula (*) preceding this example: The derivative of the left side of (†) is
Using the product rule, we see that the derivative of the far right side of (†) is
We conclude that
You Try It : Differentiate log 9 |cos x |.
You Try It : Differentiate 3 sin(3 x ) . Differentiate x sin 3 x .
Calculate the derivative of F ( x ) = x 2 · (sin x ) · 5 x .
Using formula (*), we conclude that
You Try It : Calculate ( d/dx )[(ln x ) ln x ].
Find practice problems and solutions for these concepts at: Transcendental Functions Practice Test.
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